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Does anyone know of any results that show the link between marginal cost and the output elasticities analytically? I am looking at production and cost theory books but can't find any results that posit a direct dependence.

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  • $\begingroup$ Why do you think that marginal costs are a function of the $sum$ of output elasticities? $\endgroup$
    – VARulle
    Commented Mar 25, 2021 at 13:41
  • $\begingroup$ I'm saying the sum because the sum of output elasticities = the returns to scale, which are related to marginal cost, as I say in the first sentence. $\endgroup$ Commented Mar 26, 2021 at 21:43
  • $\begingroup$ Edited the question to remove the part relating to the returns to scale, as it was confusing. $\endgroup$ Commented Apr 6, 2021 at 21:04

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I figured it out:

The first-order condition of the cost minimization problem for, say, material inputs $m_{it}$ gives:

$ \lambda \frac{\partial F}{\partial M} = P_M $

Where F is the production function, $P_M$ the material input prices. Multiply by $\frac{M}{F}$ and rearrange,

$ \lambda = \frac{P_M M}{\beta_M F} $,

where $\beta_M$ is the output elasticity with respect to material inputs. Similar results obtain for other inputs. Summing them all gives the relation to the short-run returns to scale.

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    $\begingroup$ It should be $\lambda = \frac{P_M M}{\beta_M F}$, and in the end it returns a formula for marginal cost that is not very useful. Specifically, it does not return marginal cost as a function of the sum of elasticities and therefore does not establish a connection to the returns to scale. There is no such connection, since by scaling up all inputs you typically violate cost minimization. $\endgroup$
    – VARulle
    Commented Mar 29, 2021 at 12:52
  • $\begingroup$ You're right, thanks. Typo in the formula, and one can only get the short-run returns to scale. Corrected. $\endgroup$ Commented Mar 29, 2021 at 21:10
  • $\begingroup$ I don't agree with the "not very useful", since I was mostly looking for a relation between MC and output elasticities. $\endgroup$ Commented Mar 29, 2021 at 21:11
  • $\begingroup$ Hmm, but it was formulated as if you were looking for a relation between MC and the sum of output elasticities. (That's why I asked about that on March 25.) I'm also skeptical about the "short-run returns to scale" - what's that? I still believe the formula is not very useful, since the elasticities alone don't help, as you also need to know the expansion path $M_i(F)$ to calculate MC for given output and prices. $\endgroup$
    – VARulle
    Commented Mar 30, 2021 at 11:49
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    $\begingroup$ Yeah, let's close it before the mods step in. $\endgroup$
    – VARulle
    Commented Apr 7, 2021 at 7:31

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